Tactical Planning in Healthcare Using Approximate Dynamic Programming

Tactical Planning in Healthcare using Approximate Dynamic Programming (tactisch plannen van toegangstijden in de zorg) Peter J.H. Hulshof, Martijn R.K. Mes, Richard J. Boucherie, Erwin W. Hans Center for Health care Operations Improvement and Research (CHOIR) University of Twente Friday, January 24, 2014 CHOIR Seminar University of Twente

OUTLINE Introduction Problem formulation Solution approaches Integer Linear Programming Dynamic Programming Approximate Dynamic Programming Results Conclusions This research is partly supported by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs. 2/26 CHOIR Seminar

MOTIVATION & FOCUS Motivation Long access times due to lacking match of supply and demand Varying demand (e.g., seasonality) Varying resource availability (e.g., holidays, cancellations) Limited control of serving the strategically agreed number of patients. Opportunities to improve resource utilization. Focus Integrated decision making on the tactical planning level: Patient care processes connect multiple departments and resources, which require an integrated approach. Operational decisions often depend on a tactical plan, e.g., tactical allocation of blocks of resource time to specialties and/or patient categories (master schedule / block plan). 3/26 CHOIR Seminar

CARE PROCESS & ACCESS TIME Care process: a chain of care stages for a patient, e.g., consultation, surgery, or a visit to the outpatient clinic. MRI/CT Diagnostic Services Patients going for follow-up after OR Outpatient clinic demand (patients) Outpatient clinic Operating Rooms Patients to OR Ward Patients leaving Intensive care Access time is the delay between the request for an appointment or treatment and the actual appointment or treatment. 4/26 CHOIR Seminar

OBJECTIVES OF TACTICAL PLANNING To control access times and care pathway durations To ensure quality of care for the patient and to prevent patients from seeking treatment elsewhere Decreasing care pathway duration decreases the delay between costs invested and revenues incurred To serve the strategically agreed number of patients To achieve high resource utilization and to balance workload Decreased costs and increased staff satisfaction Tactical planning requires coordinated decision making between multiple resources, multiple time periods, and multiple care pathways. 5/26 CHOIR Seminar

TACTICAL PLANNING IN OUR STUDY Typical setting: 8 care processes, 8 weeks as a planning horizon, and 4 resource types. Current way of creating/adjusting tactical plans: biweekly meeting with decision makers using spreadsheet solutions. Our objective: to provide an optimization step that supports rational decision making in tactical planning. Allocate resource capacities to care pathways, considering the variations in patient demand and resource availability. Determine a patient admission plan for each stage in every care pathway. Patient admission plan Care pathway Stage Knee Knee Hip Hip Shoulder Shoulder Week 1 5 6 2 10 4 2 Week 2 10 5 7 0 9 8 Week 3 12 10 4 7 7 5 Week 4 11 12 6 0 8 5 Week 5 9 11 8 1 8 7 Week 6 5 9 3 4 9 10 Week 7 4 5 2 4 3 8 1. Consultation 2. Surgery 1. Consultation 2. Surgery 1. Consultation 2. Surgery 6/26 CHOIR Seminar

PROBLEM FORMULATION [1/2] Discretized finite planning horizon ? 1,2, ,? Patients: Set of patient care processes ? {1,2, ,?} Each care process consists of a set of stages 1,2, ,?? A patient following care process ? follows the stages ??= ?,1 , ?,2 , ,(?,??) From now on, we denote each stage in a care process by a queue ?. Resources: Set of resource types ? {1,2, ,?} Resource capacities ??,?per resource type and time period To service a patient in queue ? requires ??,?of resource ? State: ??= (??,?,?) ?,? gives the number of patients waiting ? time units in queue ?, ?,?. 7/26 CHOIR Seminar

PROBLEM FORMULATION [2/2] After service in queue i, we have a probability ??,?that the patient is transferred to queue j. Probability to leave the system: ??,0= 1 ???,? Newly arriving patients joining queue i: ??,? Waiting list at queue ?: ??,?= ??,?,0,??,?,1, . Decision: for each time period, we determine a patient admission plan: ??,?= ??,?,0,??,?,1, , where ??,?,?indicates the number of patients to serve in time period t that have been waiting precisely u time periods at queue j. Time lag ??,?between service in i and entrance to j (might be medically required to recover from a procedure). Patients entering queue j: ??,?,0= ??,?+ ??,??? ??,?,?,? ? ? 8/26 CHOIR Seminar

ILLUSTRATION 1 queue (1 care process with 1 stage), 0/1 waiting time (2,2) (1,2) (0,2) (2,1) (1,1) States (0,1) (2,0) (1,0) (0,0) 2 3 4 0 1 Time

SOLUTION APPROACHES Mixed Integer Linear Program (MILP): Deterministic Dynamic Programming (DP): Able to incorporate uncertainty, but not scalable to realistic problem sizes Approximate Dynamic Programming (ADP): Alternative way to solve DPs, scalable to realistic problem sizes 10/26 CHOIR Seminar

ILLUSTRATION MIXED INTEGER LINEAR PROGRAM Deterministic problem - expected values: 1 arrival (2,2) (1,2) (0,2) (2,1) (1,1) States (0,1) (2,0) (1,0) (0,0) 2 3 4 0 1 Time

MIXED INTEGER LINEAR PROGRAM (MILP) Evaluate costs for all possible sequences of decisions (2,2) (1,2) (0,2) (2,1) (1,1) States (0,1) (2,0) (1,0) (0,0) 2 3 4 0 1 Time (transitions only serve as illustration, not always feasible)

MIXED INTEGER LINEAR PROGRAM (MILP) Number of patients to treat in queue j at time t with a waiting time u Number of patients in queue j at time t with waiting time u [1] Assume time lags ??,?= 1 Assume upper bound U on u Updating waiting list & bound on u Limit on the decision space 13/26 CHOIR Seminar [1] Hulshof PJ, Boucherie RJ, Hans EW, Hurink JL. (2013) Tactical resource allocation and elective patient admission planning in care processes. Health Care Manag Sci. 16(2):152-66.

PROS & CONS OF THE MILP Pros: Suitable to support integrated decision making for multiple resources, multiple time periods, and multiple patient groups. Flexible formulation (other objective functions can easily be incorporated). Cons: Quite limited in the state space. Rounding problems with fraction of patients moving from one queue to another after service. Model does not include any form of randomness. 14/26 CHOIR Seminar

DYNAMIC PROGRAMMING (DP) Calculate the exact cost-to-go backwards V1(2,2) V0(2,2) V4(2,2) V2(2,2) V3(2,2) (2,2) V4(1,2) V1(1,2) V2(1,2) V3(1,2) V0(1,2) (1,2) V4(0,2) V1(0,2) V2(0,2) V3(0,2) V0(0,2) (0,2) V4(2,1) V1(2,1) V2(2,1) V3(2,1) V0(2,1) (2,1) V4(1,1) V1(1,1) V2(1,1) V3(1,1) V0(1,1) (1,1) States V4(0,1) V1(0,1) V2(0,1) V3(0,1) V0(0,1) (0,1) V4(2,0) V1(2,0) V2(2,0) V3(2,0) V0(2,0) (2,0) V4(1,0) V1(1,0) V2(1,0) V3(1,0) V0(1,0) (1,0) V4(0,0) V1(0,0) V2(0,0) V3(0,0) V0(0,0) (0,0) 2 3 4 0 1 Time (transitions only serve as illustration, not always feasible)

DYNAMIC PROGRAMMING FORMULATION Solve: Where By backward induction. 16/26 CHOIR Seminar

THREE CURSUS OF DIMENSIONALITY 1. State space ??too large to evaluate ??(??) for all states: Suppose we have a maximum ? for the number of patients per queue and per number of time periods waiting. Then, the number of states per time period is ?? |?|. Suppose we have 40 queues (e.g., 8 care processes with an average of 5 stages), and a maximum of 4 time periods waiting. Then we have ?160states, which is intractable for any ? > 1. Decision space ??(??) (combination of patients to treat) is too large to evaluate the impact of every decision. Outcome space (possible states for the next time period) is too large to compute the expectation of cost-to-go). Outcome space is large because state space and decision space is large. 2. 3. 17/26 CHOIR Seminar

APPROXIMATE DYNAMIC PROGRAMMING (ADP) Learn cost-to-go forwards iteratively V4(2,2) V1(2,2) V2(2,2) V3(2,2) V0(2,2) (2,2) V4(1,2) V1(1,2) V2(1,2) V3(1,2) V0(1,2) (1,2) V4(0,2) V1(0,2) V2(0,2) V3(0,2) V0(0,2) (0,2) V4(2,1) V1(2,1) V1(2,1) V2(2,1) V3(2,1) V0(2,1) (2,1) V4(1,1) V1(1,1) V2(1,1) V3(1,1) V0(1,1) (1,1) States V4(0,1) V4(0,1) V1(0,1) V2(0,1) V3(0,1) V3(0,1) V0(0,1) V0(0,1) (0,1) V4(2,0) V1(2,0) V2(2,0) V3(2,0) V0(2,0) (2,0) V4(1,0) V2(1,0) V2(1,0) V1(1,0) V3(1,0) V0(1,0) (1,0) V4(0,0) V1(0,0) V2(0,0) V3(0,0) V0(0,0) (0,0) 2 3 4 0 1 Time (transitions only serve as illustration, not always feasible)

ADP FORMULATION ADP formulation uses all of the constraints from the MILP and uses a similar objective function (although formulated in a recursive manner). ADP differs from the other approaches by using sample paths. These sample paths visit one state per time period. For our problem, we are able to visit only a fraction of the states per time unit ( 1%). Remaining challenge: To design a proper approximation for the future costs ?????? That is computationally tractable. Provides a good approximation of the actual value. Is able to generalize across the state space. 19/26 CHOIR Seminar

VALUE FUNCTION APPROXIMATION [1/2] Instead of using a value for each state, we use a function ?????? that provides values for states based on particular features of the states. Examples of features: Total number of patients waiting in a queue. Average/longest waiting time of patients in a queue. Number of waiting patients requiring resource r. Combination of these. We now define the value function approximations as: ??????, ??????= ?? ? ? ? ?is a weight for each feature ? , and ??(???) is the Where ?? value of the particular feature given the state ???. 20/26 CHOIR Seminar

VALUE FUNCTION APPROXIMATION [2/2] The features can be observed as independent variables in the regression literature we use regression analysis to find the features that have a significant impact on the value function. We use the features number of patients in queue j that are u time periods waiting at time t in combination with a constant. This choice of basis functions explains a large part of the variance in the computed values with the exact DP approach (R2= 0.954). We use the recursive least squares method for non-stationary data to update the weights ?? ?. 21/26 CHOIR Seminar

EXPERIMENTS Small instances: To study convergence behavior. 8 time units, 1 resource types, 1 care process, 3 stages in the care process (3 queues), U=1 (zero or 1 time unit waiting), for DP max 8 patients per queue. 8 83 2= 2,097,152 states in total (already large for DP given that decision space and outcome space are also huge). Large instances: To study the practical relevance of our approach on real-life instances inspired by the hospitals we cooperate with. 8 time units, 4 resource types, 8 care processes, 3-7 stages per care process, U=3. 22/26 CHOIR Seminar

CONVERGENCE RESULTS ON SMALL INSTANCES Tested on 5000 random initial states. DP requires 120 hours, ADP 0.439 seconds for N=500. ADP overestimates the value functions (+2.5%) caused by the truncated state space. 120 100 80 60 40 20 0 0 50 100 150 200 250 300 350 400 450 500 DP State 1 ADP State 1 DP State 2 ADP State 2 23/26 CHOIR Seminar

PERFORMANCE ON SMALL AND LARGE INSTANCES Compare with greedy policy: fist serve the queue with the highest costs until another queue has the highest costs, or until resource capacity is insufficient. We train ADP using 100 replications after which we fix our value functions. We simulate the performance of using (i) the greedy policy and (ii) the policy determined by the value functions. We generate 5000 initial states, simulating each policy with 5000 sample paths. Results: Small instances: ADP 2% away from optimum and greedy 52% away from optimum. Large instances: ADP results 29% savings compared to greedy (higher fluctuations in resource availability or patient arrivals results in larger differences between ADP and greedy). 24/26 CHOIR Seminar

MANAGERIAL IMPLICATIONS The ADP approach can be used to establish long-term tactical plans (e.g., three month periods) in two steps: Run N iterations of the ADP algorithm to find the value functions given by the feature weights for all time periods. These value functions can be used to determine the tactical planning decision for each state and time period by generating the most likely sample path. Implementation in a rolling horizon approach: Finite horizon approach may cause unwanted and short-term focused behavior in the last time periods. Recalculation of tactical plans ensures that the most recent information is used. Recalculation can be done using the existing value function approximations and the actual state of the system. 25/26 CHOIR Seminar

QUESTIONS? Martijn Mes Assistant professor University of Twente School of Management and Governance Dept. Industrial Engineering and Business Information Systems Contact Phone: +31-534894062 Email: m.r.k.mes@utwente.nl Web: http://www.utwente.nl/mb/iebis/staff/Mes/